Array signal processing for radio-astronomy imaging and future radio telescopes

Transcription

1 Array signal processing for radio-astronomy imaging and future radio telescopes School of Engineering Bar-Ilan University Amir Leshem Faculty of EEMCS Delft University of Technology Joint wor with Alle-Jan van der Veen, Chen Ben-David, Ronny Levanda, A. Boonstra Some data and simulations were provided by S. van der Toll, S. Wijnholds, Huib Intema, Huub Rottgering and T. Clare Page 1

2 What is parametric imaging? Page 2

3 References C. Ben David and A. Leshem. Parametric high resolution techniques for radio astronomical image formation. IEEE JSTSP October A. Leshem and A-J. van der Veen. Radioastronomical imaging in the presence of strong radio interference. IEEE trans. on Information Theory. Special issue on information theoretic imaging, no. 5, pages August A. Leshem, A-J. van der Veen and A. J. Boonstra. Multichannel interference mitigation techniques in radio-astronomy. The Astrophysical Journal supplements. Volume 131, , November Page 3

4 IEEE Journal of selected topics in Signal Processing October 2008 Page 4

5 Presentation overview The matrix measurement equation Planar array Non co-planar array Polarization Calibration* Spatially varying calibration* Some ideas related to parametric imaging LS-MVI Acceleration and hardware implementation Joint imaging and calibration through SDP L1 optimization* Examples Conclusions Page 5

6 High dynamic range 1 dimensional images for simplicity Dirty images only! Theoretical dynamic range within the image 1,000,000:1 Weaest source is only 10x noise RMSE! Sources are extended off the grid One source is moved through FOV Page 6

7 High dynamic range - Array beam Page 7

8 High dynamic range - sources Page 8

9 High dynamic range - Dirty images Three 1-D dirty images: Classical, AAR and AAR with 3% RMSE calibration error on each array element in each direction Page 9

10 Imaging The correlation process T = 1 p Signals are staced into vectors x( t) x ( t),..., x ( t) Location of antenna i at time t is r ( t) Baseline between antennas i, j is r( t) r ( t) i i j Page 10

11 Imaging - the correlation process 1 R x x x x 10 s/ T s * *, E ( t) ( t ) ω = ω ω = ω( t + nts) ω( t + nts) N n= 0 ( ω ) R corresponds to the visibility V ri( t), rj( t) at frequency, i, j ω 2π Page 11

12 Coordinate system Correlations are measured only for baselines r( t) r ( t) i j Page 12

13 Imaging equation Planar array ( ) ( ) 2 2 jul vm Vf u, v A ( l, m) I f ( l, m) e π + dldm = We limit ourselves to single frequency. Alm (, ) is the "nown" amplitude response I ( l, m) is the intensity at location ( l, m). Vf of the antennas. f ( u v), is the visibility function. Page 13

14 Imaging equation: Non-coplanar array V f u, v, w ( ) 1 l = 1 m 2 2 (,, ) (, ) ( ) 2 2 jul vm wn A l m n I f l m e π + + dldm Page 14

15 The discrete measurement equation T j ( ) V ( ), ( ) ( ) l ij = V i t j t = I l e s ri rj r r s d l= 1 Page 15

16 Matrix formulation of the imaging equation Recall T j ( ) T jsr i R V = I s e = e I s e ( ) ( ) i, j d d s r r T jsr ( ) = 1 = 1 In "our" notation this translates to a where ij R = A BA H i j j I( s1 ) 0 A = a ( s1),..., a ( sd), B= 0 I( s ) d ( s) T jsr1 ( t e ) = T jsr p ( t ) e Page 16

17 The non co-planar array case Page 17

18 Matrix formulation of the imaging equation H H R = Γ( t ) ABAΓ( t ) + R t ( ) nn Page 18 Γ( t ) is a self-calibration slowly time varying pxp matrix Rnn( t ) is the noise covariance matrix. Typically AWGN but when interference exists it dominates

19 Polarimetric imaging When imaging polarized sources each element receives the two polarizations. The steering vector at epoch is now replaced by two orthogonal vectors describing circularly left and right polarization components a (), s a () s, L, R For all s, s we have a ( s ) a ( s ) 1 2, L 1, R 2 For a quasi mono-chromatic source the polarization can be described by the parameters b11 b12 b21 b 22 which are the coherence between right and left circular polarization components. Page 19

20 Polarimetric imaging The sources cross polarization coefficients b, b, b, b are connected to the Stoes parameters I,Q,U,V through the matrices b I b j 0 Q = b j 0 U b V This implies that it will be sufficient to deconvolve the parameters b, b, b, b and then recover the Stoes parameter Page 20

21 Polarimetric imaging The matrix form of the (calibrated) measurement equation now becomes (assuming N point sources) A = a, L( s1 ), a, R( s1 ),..., a, L( sn), a, R( sn) 1 1 b11 b b21 b22 B = N N b11 b12 N N b21 b 22 R = A BA H + σ 2 I, =1,...,K Page 21

22 Self calibration with polarized sources 2 R = Γ ABA Γ + σ I H H, =1,...,K ( ) Γ = diag Γ,..., Γ Γ,1, p i, ( γ ) ( ) 11 γ12 i, i, = ( γ ) ( ) 21 γ i, 22 i, Page 22

23 Deconvolution Page 23

24 Deconvolution in the (u,v) plain Clar proposed to estimate several CLEAN component in a single dirty image and subtract them Cotton-Schwab proposed to perform the subtraction in the (u,v) domain. This allows for non grid points We will always consider the Cotton-Schwab approach Perform non-grid estimation whenever possible We will discuss extension of the Clar approach Page 24

25 Deconvolution Classical Fourier imaging H I( s) = a ( s) R a ( s) D K = 1 Page 25 Super-resolution MVDR based imaging Pseudo-spectrum MVDR criterion ' wˆ ( s) = arg min I ( s) such tha t Solution w D ( ) ' H I( s) = w ( s) R w ( s) D H w ( s) a ( s) = 1 K = 1 1 wˆ ( s) = β R a s where β = K ' 1 I D( s) = a s R a s 1 ( ) ( ) H = 1 1 a s R a s 1 ( ) ( ) H

26 The adapted angular response We would lie to maintain the Gaussian receiver noise isotropic To that end we add to the MVDR equations the requirement w = 1 Still we want a solution of the form w ( ) α 1 s = R a ( s) We obtain that R a () s w = a s R a s 1 H 2 () () and the angular spectrum becomes a () s R a () s a () s R a () s K H 1 '' D s = H 2 = 1 I ( ) Page 26

27 Enhancing the image using independence Since the covariance matrices at different epochs represent indepndent observations it is reasonable to assume that the overall covariance of the data is given by MVDR R = R 1 = 1 ( ) R K T T T = 1 K Choosing a vector w for R yields w w,..., w where w = R a s 1 In this case the MVDR dirty image becomes: I ( s) = K 1 H 1 a ( s) R a ( s) Page 27

28 The AAR dirty image I ( s) AAR = K = 1 K = 1 a s R a s ( ) 1 ( ) H a s R a s ( ) 2 ( ) H Page 28

29 Dirty images (a) Classical (b) MVDR (c) AAR Page 29

30 Bandwidth synthesis Extension to bandwidth synthesis is different than the classical imaging due to non-linear structure Spectral index of sources can be included as extra parameter (not shown below) I ( s) AAR = ω K ω = 1 K = 1 a s R a s ( ) 1 ( ) H, ω, ω, ω a s R a s ( ) 2 ( ) H, ω, ω, ω Page 30

31 Classical beamforming based CLEAN H max w ( s) R w ( s) = 1 ( ) H ( ) σ 2 R = b a s a s + I n n n n= 1 H I( s) = a ( s) R a ( s) D s K N K = 1 sˆ = arg max I ( s) ˆ α = max I ( sˆ) H R = R γαˆ a ( s) a ( s) D s D Page 31

32 MVI CLEAN imaging sˆ = arg max I ( s) ˆ α = max I ( sˆ) AAR H R = R γαˆ a ( s) a ( s) s AAR Page 32

33 Least Squares power estimator to remove bias λ K 2 ( n) ( n) ( ) H ( ) = arg min n n R αa( s ) a ( s ) α =1 F Independently used in the context of multi-resolution CLEAN by Bhatnagar & Cornwell, A&A, 2004 Page 33

34 The LS-MVI algorithm Measure R,..., R Set R (0) 1 Until convergence criterion Source location estimator Power estimator Update K = R γ =0.1 R λ ( n) s = arg max I ( s) ( n) (n) AAR s K 2 ( n) ( n) ( n) H ( n) = arg min R αas ( ) a ( s ) α =1 F R R a( s ) a ( s ) ( n+ 1) ( n) ( n) ( n) H ( n) = γλ Page 34

35 Enforcing the non-negative definite constraint All source powers are non-negative and residual covariance is also non-negative 2 ( n) H ( n) R σ I αa( s ) a ( s ) 0 α 0 Solution is simple: Solve the unconstrained problem and use bisection on α finding sequentially over α that is good for all Page 35

36 Other enhancements Clar type algorithm using SDP Cotton-Schwab with off-grid estimation Page 36

37 Complexity analysis Main bottlenec: Recomputing the dirty image at each stage Complexity of naive implemntation is M 2 where M is the number of pixels in the image, p number of antennas p 2 2 K per iteration We can reduce this to M 2 pk per iteration 2 Still high compared to gridded CLEAN with 2M log 2 M operations for recomputing the dirty image, but managable. These techniques are easy to implement using parallel processors! Dedicated accelerators might be a viable option! Can use coarse grid and use local optimization to obtain accurate location estimate! We do not need intermediate images just accurate parameter estimates! Page 37

38 Robust beamforming and self calibration Assume that a C ( s) is the nominal value. ( s) is not completely nown. is a positive definite matrix describing the uncertainty at epoch a H ( a s a s ) C ( a s a s ) () () () () 1 The robust beamforming problem is given by [ ρ aˆ s aˆ s ], ( ),..., ( ) = arg max 1 subject to K 2 H H ( a a s ) C ( a a s ) 1,..., a () () 1 ρ, a R σ I ρa a 0 ρ 0 K ρ Page 38

39 Equivalent SDP formulation [ ρ aˆ s aˆ s ], ( ),..., ( ) = arg min 1 subject to ρ =1/ τ K τ, a 1,..., a K ( ()) C a a s H ( a a () s ) 1 τ 2 R σ I 0 a H 0 a τ 0 Page 39

40 Robust beamforming based self calibration K N ( ) ( ) arg min n n ˆ ( ) ( ) Γ = 1 n= 1 Γˆ = a s Γa s 2 Page 40

41 Extensions Simplified Maximum lielihood (Leshem and van der Veen 2000) Maximum lielihood using EM (Lanterman) Robust beamforming techniques, e.g., diagonal loading Generalized LS-MVI using parametric source templates (templets: shapelets, curvelets, ridgelets, wavelets ) Global optimization techniques: L1 and others Performance analysis Sensitivity to modeling assumptions Resolution limits Page 41

42 Simulated experiments -East west array with 10 elements and longest baseline 1000λ. 720 covariance matrices - Fully calibrated array Page 42

43 Resolution example Page 43

44 Extended sources example Page 44

45 Extended sources example (a) CLEAN (b) LS_MVI Page 45

46 Conclusions Exciting science can be done with future instruments Science strongly depends on technology Great challenges: Calibration is extremely challenging! Image processing can tae us beyond the equipment!! Real time adaptive interference mitigation is crucial Architectures and algorithms for extremely large arrays Comm 2-4 sensors Radar - thousands of sensors Radio astronomy 10, ,000,000 elements Page 46